The exact value of $\tan(0^\circ)$ is $0$.

To understand why $\tan(0^\circ)$ equals $0$, we first need to examine the definition of the tangent function in trigonometry. The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. This relationship can be expressed mathematically as:

When the angle $\theta$ is $0^\circ$, the opposite side of the triangle is effectively $0$ because the angle is positioned at the very start of the unit circle, lying along the x-axis. This situation implies that there is no vertical height to the triangle, rendering the length of the opposite side zero. In contrast, the adjacent side is not zero; it corresponds to the radius of the unit circle, which has a length of $1$.

Thus, for $\theta = 0^\circ$, we have:

Another perspective to consider is the unit circle itself. The unit circle is defined as a circle with a radius of $1$ centered at the origin of a coordinate plane. In the context of the unit circle, the tangent of an angle can also be understood as the ratio of the y-coordinate to the x-coordinate of the point where the terminal side of the angle intersects the circle. At $0^\circ$, this intersection point is $(1, 0)$. Therefore, we find:

This explanation demonstrates that, regardless of the approach taken, the value of $\tan(0^\circ)$ is consistently $0$.